0.1  (§15) The product topology on $X\times Y$

• $\mathrm{★}$ Def. Product topology on $X\times Y$: generated by the basis

  $$ℬ=\{U\times V\mid U\text{ open in }X,V\text{ open in }Y\}.$$

  Proof. Basis axiom (2) from $(U_1\times V_1)\cap (U_2\times V_2)=(U_1\cap U_2)\times (V_1\cap V_2)$. Note a union of two rectangles is generally not a rectangle — open sets are unions of basis elements, not basis elements.

• Thm. If $ℬ$ is a basis for $X$ and $𝒞$ for $Y$, then $𝒟=\{B\times C\mid B\in ℬ,C\in 𝒞\}$ is a basis for the product topology.

  Proof. Recognition criterion: given $(x,y)\in W$, shrink twice — first to $U\times V$, then to $B\times C$.

• Ex. Open rectangles form a basis for $ℝ\times ℝ$; this is the Euclidean topology.

• Def. Projections ${\pi }_1(x,y)=x$, ${\pi }_2(x,y)=y$.

• Lem. ${\pi }_1^{-1}(U)=U\times Y$ and ${\pi }_2^{-1}(V)=X\times V$; $(U\times Y)\cap (X\times V)=U\times V$.

• $\mathrm{★}$ Lem. Hence $𝒮=\{{\pi }_1^{-1}(U)\}\cup \{{\pi }_2^{-1}(V)\}$ is a subbasis for the product topology.

  Note. This is the form that generalises to infinite products; the rectangle picture does not.

0.2  (§16) The subspace topology

• $\mathrm{★}$ Def. For $Y\subset X$:   ${𝒯}_Y=\{Y\cap U\mid U\in 𝒯\}$. A topology on $Y$; $(Y,{𝒯}_Y)$ is a subspace.

  Proof. All three axioms come from distributivity: $\bigcup (Y\cap U_{\alpha })=Y\cap \bigcup U_{\alpha }$, likewise for finite $\cap$.

• Caution. “$V$ is open” is ambiguous when $V\subset Y\subset X$. Always say open in $Y$ or open in $X$.

• Lem. $ℬ$ a basis for $X$ $\Rightarrow$ ${ℬ}_Y=\{Y\cap B\}$ is a basis for ${𝒯}_Y$.

• Ex. $X=ℝ$, $A=[0,1)$: subspace basis consists of $[0,b)$ and $(a,b)$, . So $[0,\frac{1}{2})$ is open in $A$.

• Ex. $(X,d)$ metric, $Y\subset X$ with $d^{\prime }={d|}_{Y\times Y}$. Then $B_{d^{\prime }}(x,\epsilon )=Y\cap B_d(x,\epsilon )$ and ${𝒯}_{d^{\prime }}={({𝒯}_d)}_Y$: the subspace metric induces the subspace topology.

• Lem. $Y$ open in $X$ $\Rightarrow$ ($V\subset Y$ open in $Y$ $\iff$ open in $X$).

• $\mathrm{★}$ Thm. $A\subset X$, $B\subset Y$ subspaces. On $A\times B$ the product topology $=$ the subspace topology from $X\times Y$.

  Proof. $(A\cap U)\times (B\cap V)=(A\times B)\cap (U\times V)$ — the two bases coincide, not merely generate the same topology.

0.3  (§17) Closed sets and limit points

• Def. $A\subset X$ is closed iff $X-A$ is open.

• Ex. $[a,b]$ closed in $ℝ$. Discrete: all subsets closed. Trivial: only $\mathrm{\varnothing },X$. Cofinite: the finite sets, and $X$.

• $\mathrm{★}$ Thm. In any $X$: (1) $\mathrm{\varnothing },X$ closed; (2) arbitrary intersections of closed sets are closed; (3) finite unions of closed sets are closed.

  Proof. Pure De Morgan — the quantifiers swap. One may equally define a topology by declaring the closed sets and imposing these three.

• Caution. Not a dichotomy: sets may be both open and closed (clopen), or neither.

• Caution. As with “open”, the phrase “$A$ is closed” is ambiguous when $A\subset Y\subset X$. Say closed in $Y$ (i.e. $Y-A$ open in $Y$) or closed in $X$.

• Thm. $Y\subset X$ subspace, $A\subset Y$. Then $A$ closed in $Y$ $\iff$ $A=Y\cap B$ for some $B$ closed in $X$.

• Lem. $Y$ closed in $X$ $\Rightarrow$ ($A\subset Y$ closed in $Y$ $\iff$ closed in $X$).

• $\mathrm{★}$ Def. $\mathrm{Cl}A=\overline{A}=\bigcap \{K\text{ closed}\mid A\subset K\}$; $\mathrm{Int}A=\bigcup \{U\text{ open}\mid U\subset A\}$.

• Lem. $\overline{A}$ closed, $A\subset \overline{A}$, and $A\subset K$ closed $\Rightarrow \overline{A}\subset K$. Dually $\mathrm{Int}A$ open, $\mathrm{Int}A\subset A$, and $U\subset A$ open $\Rightarrow U\subset \mathrm{Int}A$.

• Ex. $A=[a,b)\subset ℝ$: $\overline{A}=[a,b]$, $\mathrm{Int}A=(a,b)$.

• Ex. Discrete: $\mathrm{Int}A=A=\overline{A}$. Trivial, $A$ proper nonempty: $\mathrm{Int}A=\mathrm{\varnothing }$, $\overline{A}=X$. Sierpiński ${𝒯}_a$: $\mathrm{Cl}\{a\}=X$ but $\mathrm{Cl}\{b\}=\{b\}$.

• Lem. $X-\overline{A}=\mathrm{Int}(X-A)$  and  $X-\mathrm{Int}A=\overline{X-A}$.

• Ex. $A=\mathbb{Q}\subset ℝ$: $\overline{A}=ℝ$, $\mathrm{Int}A=\mathrm{\varnothing }$ (and same for $ℝ-\mathbb{Q}$).

• $\mathrm{★}$ Thm. (Closure in a subspace.) $A\subset Y\subset X$. The closure of $A$ in $Y$ equals $Y\cap \overline{A}$, where $\overline{A}$ is the closure in $X$.

• Ex. $X=ℝ$, $Y=\mathbb{Q}$, $A=\mathbb{Q}\cap [0,\pi )$. Closure in $ℝ$ is $[0,\pi ]$; closure in $\mathbb{Q}$ is $A$ itself. So $A$ is closed in $\mathbb{Q}$.

• Def. $U$ is a neighborhood of $x$ if $x\in U$ and $U$ is open. $A$ meets $B$ if $A\cap B\ne \mathrm{\varnothing }$.

• $\mathrm{★}$ Thm. $x\in \overline{A}\iff$ every neighborhood of $x$ meets $A$. It suffices to test basis elements containing $x$.

  Proof. Contrapositive of $x\in \mathrm{Int}(X-A)$; negate twice on the board and the statement falls out.

• Ex. $A=\{1/n\mid n\in \mathbb{N}\}\subset ℝ$: every basis element $(a,b)\ni 0$ contains $(-\epsilon ,\epsilon )$, hence $1/n$ for $n>1/\epsilon$. So $0\in \overline{A}$ and $\overline{A}=\{0\}\cup A$.

  Proof. $\overline{A}$ is closed because its complement is the union of the open sets $(-\mathrm{\infty },0)$, $(1/(n+1),1/n)$ for $n\in \mathbb{N}$, and $(1,\mathrm{\infty })$.

• Def. $x$ is a limit point of $A$ if every neighborhood of $x$ meets $A-\{x\}$; equivalently $x\in \overline{A-\{x\}}$. Derived set $A^{\prime }$.

• Ex. $A=\{1/n\mid n\in \mathbb{N}\}\subset ℝ$: $A^{\prime }=\{0\}$, $\overline{A}=A\cup \{0\}$.

• $\mathrm{★}$ Thm. $\overline{A}=A\cup A^{\prime }$. Cor. $A$ closed $\iff A^{\prime }\subset A$.

• Def. $x_n\to y$: every neighborhood of $y$ contains $x_n$ for all $n\ge N$, some $N$. Enough to test basis elements.

• Caution. Ex. Sierpiński $\{a,b\}$ with ${𝒯}_a$: the constant sequence $(a,a,\mathrm{\dots })$ converges to both $a$ and $b$. Limits need not be unique.

• $\mathrm{★}$ Def. $X$ is Hausdorff if for all $x\ne y$ there are neighborhoods $U\ni x$, $V\ni y$ with $U\cap V=\mathrm{\varnothing }$.

• Ex. $\{a,b\}$ discrete is Hausdorff: $\{a\},\{b\}$ are disjoint neighborhoods. $\{a,b\}$ with ${𝒯}_a$ is not: the only neighborhood of $b$ is $X$, which meets every neighborhood of $a$.

• Lem. Every metric space is Hausdorff

  Proof. take $\epsilon =d(x,y)/2$ and use the triangle inequality.

• Rmk. Hausdorff is inherited by finer topologies — “enough open sets locally”.

• $\mathrm{★}$ Thm. $X$ Hausdorff $\Rightarrow$ every convergent sequence has exactly one limit; write $y={lim}_{n\to \mathrm{\infty }}{x}_n$.

  Proof. If $x_n\to y$ and $x_n\to z$ with $y\ne z$, then $x_n\in U\cap V=\mathrm{\varnothing }$ for large $n$.

• Thm. $X$ Hausdorff $\Rightarrow$ every finite subset is closed.

  Proof. Enough for $\{x\}$: any $y\ne x$ has $V\ni y$ missing $x$.

• Thm. $X$ Hausdorff, $A\subset X$: $x\in A^{\prime }$ $\iff$ every neighborhood of $x$ meets $A$ in infinitely many points.

• Lem. $X,Y$ Hausdorff $\Rightarrow X\times Y$ Hausdorff.